Mathematische Zeitschrift

, Volume 292, Issue 1–2, pp 571–589 | Cite as

Conformally Einstein–Maxwell Kähler metrics and structure of the automorphism group

  • Akito FutakiEmail author
  • Hajime Ono


Let (Mg) be a compact Kähler manifold and f a positive smooth function such that its Hamiltonian vector field \(K = J\mathrm {grad}_g f\) for the Kähler form \(\omega _g\) is a holomorphic Killing vector field. We say that the pair (gf) is conformally Einstein–Maxwell Kähler metric if the conformal metric \(\tilde{g} = f^{-2}g\) has constant scalar curvature. In this paper we prove a reductiveness result of the reduced Lie algebra of holomorphic vector fields for conformally Einstein–Maxwell Kähler manifolds, extending the Lichnerowicz–Matsushima Theorem for constant scalar curvature Kähler manifolds. More generally we consider extensions of Calabi functional and extremal Kähler metrics, and prove an extension of Calabi’s theorem on the structure of the Lie algebra of holomorphic vector fields for extremal Kähler manifolds. The proof uses a Hessian formula for the Calabi functional under the set up of Donaldson-Fujiki picture.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan
  2. 2.Yau Mathematical Sciences CenterTsinghua UniversityBeijingChina
  3. 3.Department of MathematicsSaitama UniversitySaitamaJapan

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